Ladder equation for the three-particle vertex and its approximate solution

TL;DR

This paper extends the Bethe-Salpeter equations to three-particle ladders, proposing an approximation for the three-particle vertex and analyzing its accuracy for weak interactions.

Contribution

It introduces a generalized ladder equation for three-particle vertices and an approximation method using two-particle irreducible vertices, providing insights into their accuracy.

Findings

Approximation is accurate only for weak interactions.

The method provides qualitative insights into non-linear response functions.

Exact three-particle vertex can be obtained if the irreducible vertex is known.

Abstract

We generalize the three two-particle Bethe-Salpeter equations to ten three-particle ladders. These equations are exact and yield the exact three-particle vertex, if we knew the three-particle vertex irreducible in one of the ten channels. However, as we do not have this three-particle irreducible vertex at hand, we approximate this building block for the ladder by the sum of two-particle irreducible vertices each connecting two fermionic lines. The comparison to the exact solution shows that this approximation is only good for rather weak interactions and even than only qualitatively - at least for the non-linear response function analyzed.